Integral of $x \cos\left(x\right)$

Q: What is the integral of x * cos(x)?

The integral of x ≤ft(x) is ≤ft(x) + x ≤ft(x) + C.

The integral of $x \cos\left(x\right)$ is $\cos\left(x\right) + x \sin\left(x\right) + C$ .

\int x \cos\left(x\right)\,dx = \cos\left(x\right) + x \sin\left(x\right) + C

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Graph of f(x) = x * cos(x) with the area from 0 to π/2 shaded. The shaded area equals the definite integral, about 0.57079633.

How to work it out

1
We want an antiderivative of f(x) = x * cos(x).
$\int x \cos\left(x\right)\,dx$
2
Method: integration by parts with u = x, dv = cos x dx.
3
That gives the antiderivative; add the constant C because any constant disappears when you differentiate:
$\int x \cos\left(x\right)\,dx = \cos\left(x\right) + x \sin\left(x\right) + C$
4
Check by differentiating the answer; you must get the original function back:
$\frac{d}{dx}\left[\cos\left(x\right) + x \sin\left(x\right)\right] = -\sin\left(x\right) + \sin\left(x\right) + x \cos\left(x\right)$
5
As a definite integral, evaluate the antiderivative at both ends and subtract. Between 0 and π/2:
$\int_{0}^{\frac{\pi}{2}} x \cos\left(x\right)\,dx = \Big[\cos\left(x\right) + x \sin\left(x\right)\Big]_{0}^{\frac{\pi}{2}} \approx 0.57079633$

LaTeX source of the formulas on this page

\[ \int x \cos\left(x\right)\,dx = \cos\left(x\right) + x \sin\left(x\right) + C \]
\[ \int x \cos\left(x\right)\,dx \]
\[ \frac{d}{dx}\left[\cos\left(x\right) + x \sin\left(x\right)\right] = -\sin\left(x\right) + \sin\left(x\right) + x \cos\left(x\right) \]
\[ \int_{0}^{\frac{\pi}{2}} x \cos\left(x\right)\,dx = \Big[\cos\left(x\right) + x \sin\left(x\right)\Big]_{0}^{\frac{\pi}{2}} \approx 0.57079633 \]

Definite integrals of the same function

Interval	Value
from 0 to π/2	0.57079633
from 0 to 0.7853981633974483	0.26246715

Questions

What is the integral of x * cos(x)?

The integral of $x \cos\left(x\right)$ is $\cos\left(x\right) + x \sin\left(x\right) + C$ .

Why do we add + C?

Every function F(x) + C has the same derivative, so the antiderivative is only defined up to a constant.

What is the integral of x * cos(x) from 0 to π/2?

About 0.57079633.

Integral of xcos⁡(x)x \cos\left(x\right)xcos(x)

How to work it out

Definite integrals of the same function

Questions

More integrals

Integral of $x \cos\left(x\right)$